Derivative of trace of log of matrix products w.r.t. a matrix

Question

Is there a way to calculate $$\frac{\partial\; \mbox{tr}\{\log(X^tBX)\}}{\partial X},$$ where $X$ and $B$ are $n\times n$ matrices?

Answer 1

$${\rm tr}\log (X^t B X)=\log{\rm det}\,(X^t BX)=2\log{\rm det}\,X+\log{\rm det}\,B$$ $$\Rightarrow\frac{\partial}{\partial X}{\rm tr}\log (X^t B X)=2X^{-1},$$ in view of Jacobi's formula